The probability that the sum of two integers m and n , where m, n 1, 2, , 50 , chosen randomly and independently, being divisible by 3 is

Step-by-step solution

Let the given set of 50 integers be partitioned into three subsets S₀, S₁, S₂ based on the remainder when divided by 3 . The number of elements in these subsets are n(S₀) = 16 (multiples of 3 ), n(S₁) = 17 (leaving remainder 1 ), and n(S₂) = 17 (leaving remainder 2 ). For the sum m+n to be a multiple of 3 , the pair (m, n) must be selected from (S₀, S₀) , (S₁, S₂) , or (S₂, S₁) . The number of favorable ways to choose the pair (m, n) is 16 16 + 17 17 + 17 17 = 256 + 289 + 289 = 834 . Since m and n are chosen indepe